6.2
The Natural Base e
Essential Question
What is the natural base e?
So far in your study of mathematics, you have worked with special numbers such as
π and i. Another special number is called the natural base and is denoted by e. The
natural base e is irrational, so you cannot find its exact value.
Approximating the Natural Base e
Work with a partner. One way to approximate the natural base e is to approximate
the sum
1
1
1
1
1 + — + — + — + —— + . . . .
1 1 2 1 2 3 1 2 3 4
⋅
USING TOOLS
STRATEGICALLY
To be proficient in
math, you need to use
technological tools to
explore and deepen
your understanding
of concepts.
⋅ ⋅
⋅ ⋅ ⋅
Use a spreadsheet or a graphing calculator to approximate this sum. Explain the steps
you used. How many decimal places did you use in your approximation?
Approximating the Natural Base e
Work with a partner. Another way to approximate the natural base e is to consider
the expression
x
( 1 + 1x ) .
—
As x increases, the value of this expression approaches the value of e. Copy and
complete the table. Then use the results in the table to approximate e. Compare this
approximation to the one you obtained in Exploration 1.
x
(
1
1+—
x
101
102
103
104
105
106
x
)
Graphing a Natural Base Function
Work with a partner. Use your approximate value of e in Exploration 1 or 2 to
complete the table. Then sketch the graph of the natural base exponential function
y = e x. You can use a graphing calculator and the e x key to check your graph.
What are the domain and range of y = e x? Justify your answers.
x
−2
−1
0
1
2
y = ex
Communicate Your Answer
4. What is the natural base e?
5. Repeat Exploration 3 for the natural base exponential function y = e−x. Then
compare the graph of y = e x to the graph of y = e−x.
6. The natural base e is used in a wide variety of real-life applications. Use the
Internet or some other reference to research some of the real-life applications of e.
Section 6.2
hsnb_alg2_pe_0602.indd 303
The Natural Base e
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6.2 Lesson
What You Will Learn
Define and use the natural base e.
Graph natural base functions.
Core Vocabul
Vocabulary
larry
Solve real-life problems.
natural base e, p. 304
The Natural Base e
Previous
irrational number
properties of exponents
percent increase
percent decrease
compound interest
The history of mathematics is marked by the discovery
of special numbers, such as π and i. Another special
number is denoted by the letter e. The number is called the
x
1
natural base e. The expression 1 + — approaches e as
x
x increases, as shown in the graph and table.
(
)
y
2
(
1 x
8
12
y= 1+x
1
0
x
y=e
3
0
4
(
101
102
103
104
105
106
2.59374
2.70481
2.71692
2.71815
2.71827
2.71828
x
x
( 1 + —x1 )
Core Concept
The Natural Base e
The natural base e is irrational. It is defined as follows:
(
x
)
1
As x approaches +∞, 1 + — approaches e ≈ 2.71828182846.
x
Simplifying Natural Base Expressions
Check
Simplify each expression.
You can use a calculator to check
the equivalence of numerical
expressions involving e.
a. e3
e^(3)*e^(6)
8103.083928
e^(9)
8103.083928
⋅e
6
16e5
b. —
4e4
c. (3e−4x)2
16e5
b. —
= 4e5 − 4
4e4
c. (3e−4x)2 = 32(e−4x)2
SOLUTION
⋅
a. e3 e6 = e3 + 6
= e9
= 4e
= 9e−8x
9
=—
e8x
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Simplify the expression.
1. e7
304
Chapter 6
hsnb_alg2_pe_0602.indd 304
⋅e
4
24e8
8e
2. —
5
3. (10e−3x)3
Exponential and Logarithmic Functions
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Graphing Natural Base Functions
Core Concept
Natural Base Functions
A function of the form y = aerx is called a natural base exponential function.
• When a > 0 and r > 0, the function is an exponential growth function.
• When a > 0 and r < 0, the function is an exponential decay function.
The graphs of the basic functions y = e x and y = e−x are shown.
7
y
7
exponential
5
growth
3
exponential
decay
5
y = ex
y = e−x
(1, 2.718)
(0, 1)
−4
y
3
(1, 0.368)
(0, 1)
−2
2
4
x
−4
−2
2
x
4
Graphing Natural Base Functions
Tell whether each function represents exponential growth or exponential decay.
Then graph the function.
b. f (x) = e−0.5x
a. y = 3ex
LOOKING FOR
STRUCTURE
You can rewrite natural
base exponential functions
to find percent rates of
change. In Example 2(b),
f (x) = e−0.5x
= (e−0.5)x
SOLUTION
a. Because a = 3 is positive and
r = 1 is positive, the function is
an exponential growth function.
Use a table to graph the function.
b. Because a = 1 is positive and
r = −0.5 is negative, the function
is an exponential decay function.
Use a table to graph the function.
x
−2
−1
0
1
x
−4
−2
0
2
y
0.41 1.10
3
8.15
y
7.39 2.72
1
0.37
≈ (0.6065)x
= (1 − 0.3935)x.
16
So, the percent decrease is
about 39.35%.
y
y
(−4, 7.39)
12
(−1, 1.10)
8
6
(1, 8.15)
4
(−2, 2.72)
(−2, 0.41)
−4
−2
(0, 3)
2
2
(2, 0.37)
(0, 1)
4x
Monitoring Progress
−4
−2
2
4x
Help in English and Spanish at BigIdeasMath.com
Tell whether the function represents exponential growth or exponential decay.
Then graph the function.
1
4. y = —2 e x
5. y = 4e−x
Section 6.2
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6. f (x) = 2e2x
The Natural Base e
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Solving Real-Life Problems
You have learned that the balance of an account earning compound interest is given by
r nt
A = P 1 + — . As the frequency n of compounding approaches positive infinity, the
n
compound interest formula approximates the following formula.
)
(
Core Concept
Continuously Compounded Interest
When interest is compounded continuously, the amount A in an account after
t years is given by the formula
A = Pe rt
where P is the principal and r is the annual interest rate expressed as a decimal.
Modeling with Mathematics
12,000
You and your friend each have accounts that earn annual interest compounded
continuously. The balance A (in dollars) of your account after t years can be modeled
by A = 4500e0.04t. The graph shows the balance of your friend’s account over time.
Which account has a greater principal? Which has a greater balance after 10 years?
10,000
SOLUTION
Your Friend’s Account
Balance (dollars)
A
1. Understand the Problem You are given a graph and an equation that represent
account balances. You are asked to identify the account with the greater principal
and the account with the greater balance after 10 years.
8,000
6,000
4,000
2. Make a Plan Use the equation to find your principal and account balance after
10 years. Then compare these values to the graph of your friend’s account.
(0, 4000)
2,000
0
0
4
8
12
16
Year
t
3. Solve the Problem The equation A = 4500e0.04t is of the form A = Pe rt, where
P = 4500. So, your principal is $4500. Your balance A when t = 10 is
A = 4500e0.04(10) = $6713.21.
Because the graph passes through (0, 4000), your friend’s principal is $4000. The
graph also shows that the balance is about $7250 when t = 10.
So, your account has a greater principal, but your friend’s account has a
greater balance after 10 years.
MAKING
CONJECTURES
You can also use this
reasoning to conclude that
your friend’s account has
a greater annual interest
rate than your account.
4. Look Back Because your friend’s account has a lesser principal but a greater
balance after 10 years, the average rate of change from t = 0 to t = 10 should be
greater for your friend’s account than for your account.
A(10) − A(0) 6713.21 − 4500
Your account: —— = —— = 221.321
10 − 0
10
A(10) − A(0) 7250 − 4000
Your friend’s account: —— ≈ —— = 325
10 − 0
10
Monitoring Progress
✓
Help in English and Spanish at BigIdeasMath.com
7. You deposit $4250 in an account that earns 5% annual interest compounded
continuously. Compare the balance after 10 years with the accounts in Example 3.
306
Chapter 6
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Exponential and Logarithmic Functions
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6.2
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. VOCABULARY What is the natural base e?
1
2. WRITING Tell whether the function f (x) = —3 e 4x represents exponential growth or exponential decay.
Explain.
Monitoring Progress and Modeling with Mathematics
In Exercises 3–12, simplify the expression.
(See Example 1.)
⋅
⋅
4. e−4 e6
3. e3 e5
11e9
22e
27e7
3e
5. —
10
6. —
4
7. (5e7x)4
8. (4e−2x)3
11.
⋅ ⋅
24. y = e−2x
25. y = 4e−0.5x
26. y = 0.75e x
A.
e8
12.
⋅ ⋅
ex
e4
(−1, 7.39)
14.
✗
✗
y
8
(−1, 6.59)
6
(0, 4)
ex + 3
1
ERROR ANALYSIS In Exercises 13 and 14, describe and
correct the error in simplifying the expression.
13.
B.
y
8
10. √ 8e12x
9. √ 9e6x
e−6x
23. y = e2x
3—
—
ex
ANALYZING EQUATIONS In Exercises 23–26, match the
function with its graph. Explain your reasoning.
−4
−2
C.
= 4e 6x
e5x
−2
D.
y
4
(1, 7.39)
2
(1, 2.04)
2
y
8
4
−2
4x
2
6
(0, 0.75)
−4
−4
4x
6
2
= e 5x − 2x
—
e−2x
= e 3x
2
8
(4e3x)2 = 4e(3x)(2)
2
(0, 1)
(0, 1)
4x
−4
−2
4x
2
USING STRUCTURE In Exercises 27–30, use the
In Exercises 15–22, tell whether the function represents
exponential growth or exponential decay. Then graph the
function. (See Example 2.)
properties of exponents to rewrite the function in
the form y = a(1 + r) t or y = a(1 − r) t. Then find the
percent rate of change.
15. y = e3x
16. y = e−2x
27. y = e−0.25t
28. y = e−0.75t
17. y = 2e−x
18. y = 3e2x
29. y = 2e0.4t
30. y = 0.5e0.8t
19. y = 0.5e x
20. y = 0.25e−3x
USING TOOLS In Exercises 31–34, use a table of values
21. y = 0.4e−0.25x
22. y = 0.6e0.5x
or a graphing calculator to graph the function. Then
identify the domain and range.
31. y = e x − 2
32. y = e x + 1
33. y = 2e x + 1
34. y = 3e x − 5
Section 6.2
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The Natural Base e
307
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accounts for a house and education earn annual
interest compounded continuously. The balance H
(in dollars) of the house fund after t years can be
modeled by H = 3224e0.05t. The graph shows the
balance in the education fund over time. Which
account has the greater principal? Which account has
a greater balance after 10 years? (See Example 3.)
approximates A = Pe rt as n approaches positive
infinity.
of two integers? Explain.
40. MAKING AN ARGUMENT Your friend evaluates
f (x) = e−x when x = 1000 and concludes that the
graph of y = f (x) has an x-intercept at (1000, 0).
Is your friend correct? Explain your reasoning.
H
10,000
Balance (dollars)
nt
)
39. WRITING Can the natural base e be written as a ratio
Education Account
41. DRAWING CONCLUSIONS You invest $2500 in an
8,000
account to save for college. Account 1 pays 6%
annual interest compounded quarterly. Account 2 pays
4% annual interest compounded continuously. Which
account should you choose to obtain the greater
amount in 10 years? Justify your answer.
6,000
(0, 4856)
4,000
2,000
0
r
n
(
38. THOUGHT PROVOKING Explain why A = P 1 + —
35. MODELING WITH MATHEMATICS Investment
0
4
8
12
16
t
42. HOW DO YOU SEE IT? Use the graph to complete
Year
each statement.
a. f (x) approaches ____
as x approaches +∞.
36. MODELING WITH MATHEMATICS Tritium and
sodium-22 decay over time. In a sample of tritium,
the amount y (in milligrams) remaining after t years is
given by y = 10e−0.0562t. The graph shows the amount
of sodium-22 in a sample over time. Which sample
started with a greater amount? Which has a greater
amount after 10 years?
b. f (x) approaches ____
as x approaches −∞.
Amount
(milligrams)
Sodium-22 Decay
f
x
43. PROBLEM SOLVING The growth of Mycobacterium
y
20
tuberculosis bacteria can be modeled by the function
N(t) = ae 0.166t, where N is the number of cells after
t hours and a is the number of cells when t = 0.
10
0
y
0
10
a. At 1:00 p.m., there are 30 M. tuberculosis bacteria
in a sample. Write a function that gives the number
of bacteria after 1:00 p.m.
20 t
Year
b. Use a graphing calculator to graph the function in
part (a).
37. OPEN-ENDED Find values of a, b, r, and q such that
f (x) = aerx and g(x) = be qx are exponential decay
f (x)
functions, but — represents exponential growth.
g(x)
c. Describe how to find the number of cells in the
sample at 3:45 p.m.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Write the number in scientific notation. (Skills Review Handbook)
44. 0.006
45. 5000
46. 26,000,000
47. 0.000000047
Find the inverse of the function. Then graph the function and its inverse. (Section 5.6)
48. y = 3x + 5
—
50. y = √ x + 6
308
Chapter 6
hsnb_alg2_pe_0602.indd 308
49. y = x2 − 1, x ≤ 0
51. y = x3 − 2
Exponential and Logarithmic Functions
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